Let $C_1,C_2,C_3,\mathrm{......}$ be the circles with radius $r_1$ such that equation of circle $C_1:x^2+y^2=t_1^2.$ Circle $C_1$ intersects positive x-axis and y-axis at $A_1$ and $B_1$ respectively. There exist a point $P_i,$ perpendicular from $P_i$ on $O{A}_i$ and $O{B}_i$ are points $D_i$ and $E_i$ respectively such that  $O{D}_i:O{E}_i=3:4$ and there exist a point $Q_i$ on circle $C_{i+1,}$ if  $A_i{Q}_i=P_i{Q}_i$ and $\angle A_i{Q}_i{P}_i=\frac{\pi }{2}.$ If $S_i$ be the area of triangle $A_i{Q}_i{P}_i$ and $r_1>r_2>r_3>\mathrm{.....}>r_{n,}$ where $r_1=1,$ then $\underset{n\to \infty }{\lim }\left(\sum _{i=1}^n{S}_i\right)$ is less than or equal to :